Find an extremal of functional $ {\rm J}\left[(x\left(t\right),y\left(t\right)\right] =\int_{0}^{\pi/4}2y - 4x^{2} + x'^{\,2} - y'^{\,2}\,{\rm d}t $?

Question

How to find an extremal of the functional $$ {\rm J}\left[(x\left(t\right),y\left(t\right)\right] =\int_{0}^{\pi/4}\left[2y - 4x^{2} + x'^{\,2} - y'^{\,2}\right]\,{\rm d}t\ {\large ?} $$ I don't know what is the Euler-Lagrange equation in the case when a functional depends on two variables.

Answer 1

The Euler-Lagrange equations are going to be $$ \frac{\mathrm{d}}{\mathrm{d}t} \frac{\partial \mathcal{L}}{\partial x^\prime} = \frac{\partial \mathcal{L}}{\partial x} \qquad \frac{\mathrm{d}}{\mathrm{d}t} \frac{\partial \mathcal{L}}{\partial y^\prime} = \frac{\partial \mathcal{L}}{\partial y} $$ that is $$ 2 x^{\prime\prime} = - 8 x \qquad -2 y^{\prime\prime} = 2 $$ with boundary conditions $x(0) = x_0$ and $x(\pi/4) = x_1$ and $y(0)=y_0$ and $y(\pi/4)=y_1$.

Finding the solution is straightforward.